Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes

TL;DR

This paper derives Fokker-Planck equations for nonlinear stochastic systems driven by non-Gaussian Levy processes, providing a series expansion approach to handle the complexity of the associated operators.

Contribution

It introduces a novel series-based method to explicitly formulate Fokker-Planck equations for systems with non-Gaussian Levy noise, overcoming previous analytical difficulties.

Findings

Derived infinite series forms of Fokker-Planck equations

Applied the method to several example systems

Demonstrated the approach's effectiveness in complex cases

Abstract

The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian L\'evy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian L\'evy processes. A few examples are presented to illustrate the method.